Book: Noncommutative Functional Calculus, Colombo et al


Noncommutative Functional Calculus
Theory and Applications of Slice Hyperholomorphic Functions

Series: Progress in Mathematics, Vol. 289

Authors: Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa

This book presents a functional calculus for n-tuples of not necessarily commuting linear operators. In particular, a functional calculus for quaternionic linear operators is developed. These calculi are based on a new theory of hyperholomorphicity for functions with values in a Clifford algebra: the so-called slice monogenic functions which are carefully described in the book. In the case of functions with values in the algebra of quaternions these functions are named slice regular functions.

Except for the appendix and the introduction all results are new and appear for the first time organized in a monograph. The material has been carefully prepared to be as self-contained as possible. The intended audience consists of researchers, graduate and postgraduate students interested in operator theory, spectral theory,  hypercomplex analysis, and mathematical physics.

http://www.springer.com/mathematics/analysis/book/978-3-0348-0109-6

 

Table of Contents
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Plan of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Slice monogenic functions 17
2.1 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Slice monogenic functions: definition and properties . . . . . . . . . 23
2.3 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Cauchy integral formula, I . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Zeros of slicemonogenic functions . . . . . . . . . . . . . . . . . . 42
2.6 The slicemonogenic product . . . . . . . . . . . . . . . . . . . . . 47
2.7 Slice monogenic Cauchy kernel . . . . . . . . . . . . . . . . . . . . 53
2.8 Cauchy integral formula, II . . . . . . . . . . . . . . . . . . . . . . 60
2.9 Duality Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.10 TopologicalDuality Theorems . . . . . . . . . . . . . . . . . . . . . 73
2.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3 Functional calculus for n-tuples of operators 81
3.1 The S-resolvent operator and the S-spectrum . . . . . . . . . . . . 82
3.2 Properties of the S-spectrum . . . . . . . . . . . . . . . . . . . . . 86
3.3 The functional calculus . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4 Algebraic rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5 The spectral mapping and the S-spectral radius theorems . . . . . 93
3.6 Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.7 Functional calculus for unbounded operators and algebraic properties101
3.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4 Quaternionic Functional Calculus 113
4.1 Notation and definition of slice regular functions . . . . . . . . . . 113
4.2 Properties of slice regular functions . . . . . . . . . . . . . . . . . . 117
4.3 Representation Formula for slice regular functions . . . . . . . . . 121
4.4 The slice regular Cauchy kernel . . . . . . . . . . . . . . . . . . . . 129
4.5 The Cauchy integral formula II . . . . . . . . . . . . . . . . . . . . 134
4.6 Linear bounded quaternionic operators . . . . . . . . . . . . . . . . 136
4.7 The S-resolvent operator series . . . . . . . . . . . . . . . . . . . . 138
4.8 The S-spectrum and the S-resolvent operators . . . . . . . . . . . 141
4.9 Examples of S-spectra . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.10 The quaternionic functional calculus . . . . . . . . . . . . . . . . . 146
4.11 Algebraic properties of the quaternionic functional calculus . . . . 151
4.12 The S-spectral radius . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.13 The S-spectral mapping and the composition theorems . . . . . . . 156
4.14 Bounded perturbations of the S-resolvent operator . . . . . . . . . 159
4.15 Linear closed quaternionic operators . . . . . . . . . . . . . . . . . 166
4.16 The functional calculus for unbounded operators . . . . . . . . . . 173
4.17 An application: uniformly continuous quaternionic semigroups . . . 180
4.18 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5 Appendix: The Riesz–Dunford functional calculus 201
5.1 Vector-valued functions of a complex variable . . . . . . . . . . . . 201
5.2 The functional calculus for linear bounded operators . . . . . . . . 203
5.3 The functional calculus for unbounded operators . . . . . . . . . . 208
Bibliography 211
Index 219

 

Source: Email by I. Sabadini, 23 Mar. 2011, 2:12am, i.sabadiniATalice.it

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